Question

Let A and B be any two $3 \times 3$ matrices. If A is symmetric and B is skew symmetric, then the matrix $AB - BA$ is:
A) Skew symmetric
B) Symmetric
C) Neither symmetric nor skew symmetric
D) I or –I, where I is the identity matrix

Answer 1

Here we will find the transpose of the given matrix and then use the concept of symmetric and skew symmetric matrix i.e.
If a matrix X is symmetric then ${\left( X \right)^T} = X$
If a matrix Y is skew symmetric then ${\left( Y \right)^T} = - Y$

Complete step-by-step answer:
The given matrix is:-
$AB - BA$
Taking transpose of the above matrix we get:-
${\left( {AB - BA} \right)^T} = {\left( {AB} \right)^T} - {\left( {BA} \right)^T}$
Now we know that:
${\left( {XY} \right)^T} = {Y^T}{X^T}$
Hence, applying this property we get:-
${\left( {AB - BA} \right)^T} = {B^T}{A^T} - {A^T}{B^T}$……………………………………….(1)
Now since A is symmetric matrix
Therefore, ${A^T} = A$
Since B is skew symmetric matrix
Therefore,
${B^T} = - B$
Hence substituting the values in equation 1 we get:-
${\left( {AB - BA} \right)^T} = \left( { - B} \right)\left( A \right) - \left( A \right)\left( { - B} \right)$
Simplifying it further we get:-

{\left( {AB - BA} \right)^T} = - BA + AB \\\ \Rightarrow {\left( {AB - BA} \right)^T} = AB - BA \\\ \end{gathered} $$ Hence, $$AB - BA$$ is a symmetric matrix. **Therefore, option A is the correct option.** **Note:** Students should note that only the square matrices can be symmetric or skew-symmetric form. Also, matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A and the upper triangular matrix is equal to the lower triangular matrix $$\left[ {\begin{array}{*{20}{c}} a&b;&c; \\\ b&d;&f; \\\ c&f;&e; \end{array}} \right]$$ Matrix A is said to be skew-symmetric if the transpose of matrix A is equal to negative of matrix A and the upper triangular matrix is negative to the lower triangular matrix or vice-versa. $$\left[ {\begin{array}{*{20}{c}} a&b;&c; \\\ { - b}&d;&f; \\\ { - c}&{ - f}&e; \end{array}} \right]$$